Question

Apresente o estudo dos sinais das funções a seguir: (anexar uma foto da atividade)

a) f(x) = x2 - 7x + 10;

b) f(x) = - x2 + 5x - 4

Answer 1

Explicação passo-a-passo:

a) $\sf f(x)=x^2-7x+10$

=> Raízes

$\sf x^2-7x+10=0$

$\sf \Delta=(-7)^2-4\cdot1\cdot10$

$\sf \Delta=49-40$

$\sf \Delta=9$

$\sf x=\dfrac{-(-7)\pm\sqrt{9}}{2\cdot1}=\dfrac{7\pm3}{2}$

$\sf x'=\dfrac{7+3}{2}~\Rightarrow~x'=\dfrac{10}{2}~\Rightarrow~\red{x'=5}$

$\sf x"=\dfrac{7-3}{2}~\Rightarrow~x"=\dfrac{4}{2}~\Rightarrow~\red{x"=2}$

As raízes são 5 e 2

=> Estudo dos sinais

• $\sf f(x) > 0,~para~x < 2~ou~x > 5$

• $\sf f(x) < 0,~para~2 < x < 5$

• $\sf f(x)=0,~para~x=2~ou~x=5$

b) $\sf f(x)=-x^2+5x-4$

=> Raízes

$\sf -x^2+5x-4=0$

$\sf \Delta=5^2-4\cdot(-1)\cdot4$

$\sf \Delta=25-16$

$\sf \Delta=9$

$\sf x=\dfrac{-5\pm\sqrt{9}}{2\cdot(-1)}=\dfrac{-5\pm3}{-2}$

$\sf x'=\dfrac{-5+3}{-2}~\Rightarrow~x'=\dfrac{-2}{-2}~\Rightarrow~\red{x'=1}$

$\sf x"=\dfrac{-5-3}{-2}~\Rightarrow~x"=\dfrac{-8}{-2}~\Rightarrow~\red{x"=4}$

As raízes são 1 e 4

=> Estudo dos sinais

• $\sf f(x) > 0,~para~1 < x < 4$

• $\sf f(x) < 0,~para~x < 1~ou~x > 4$

• $\sf f(x)=0,~para~x=1~ou~x=4$